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wdlang
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why group SO(3) is not
any good reference on the relation of SU(2) and SO(3)?
any good reference on the relation of SU(2) and SO(3)?
Fredrik said:Write down the most general complex 2×2 matrix U and find out what relationships between its components you can derive from the conditions [itex]U^\dagger U=I[/itex] and [itex]\det U=1[/itex]. You should end up with the condition that defines the unit 3-sphere.
Fredrik said:The first thing you should know is that any 2x2 traceless self-adjoint matrix can be written as
[tex]\begin{pmatrix}x_3 & x_1-ix_2\\ x_1+ix_2 & -x_3\end{pmatrix}=x_i\sigma_i[/tex]
so the set of Pauli spin matrices is just a basis of the (real) vector space of (complex) 2x2 traceless self-adjoint matrices.
Fredrik said:You can prove that SU(2) is homeomorphic to a 3-sphere, like this:
The relationship between SO(3) and SU(2) can be found by first noting that [tex]\mathbb R^3[/tex] is isomorphic to the 3-dimensional real vector space of complex 2×2 traceless self-adjoint matrices, and then showing that if X is a member of that space, and U is a member of SU(2), then
[tex]X\mapsto UXU^\dagger[/tex]
is a proper rotation, i.e. a member of SO(3). Since you can change the sign of the U without changing the result, there are two members of SU(2) for each member of SO(3).
Not sure what the best reference is if you don't want to figure out the details for yourself. I think Weinberg's QFT book covers this pretty well (vol. 1, chapter 2), but he's actually doing it to find the relationship between SO(3,1) and SL(2,C), so he's doing essentially the same thing with the "traceless" condition dropped, and U not necessarily unitary. This brings a fourth basis vector into the picture: the 2×2 identity matrix.
In group theory, the concept of simply connectedness refers to a group being topologically connected and having no "holes" or "gaps". In the case of SU(2), being simply connected means that any loop in the group can be continuously shrunk to a point without leaving the group. This has important implications in mathematical physics and the study of symmetry, as it allows for a unique and well-defined path between any two elements in the group.
The proof of SU(2) being simply connected involves showing that any closed loop in the group can be continuously deformed to a point. This can be done using the concept of homotopy, which is a mathematical tool for studying continuous deformations. By constructing a homotopy between any two elements in SU(2), it can be shown that the group is simply connected.
One way to visualize SU(2) is as the group of unit quaternions, which can be represented as points on a 3-sphere in 4 dimensions. This is known as the Hopf fibration, and it provides a geometric understanding of the group's structure and its relationship to the 3-sphere. Additionally, SU(2) is the double cover of the 3-dimensional rotation group, SO(3), which also has a close connection to the 3-sphere.
In physics, SU(2) is a fundamental group used to describe the symmetries of physical systems. This includes the symmetries of particles in quantum mechanics, as well as the symmetries of the laws of nature in general relativity. The simply connectedness of SU(2) ensures that these symmetries are well-defined and allows for a deeper understanding of the underlying mathematical structure of physical theories.
Yes, there are several other important mathematical properties of SU(2) that make it a fascinating and useful group in physics and mathematics. These include being a compact Lie group, having a non-trivial center, and being a special unitary group. Additionally, SU(2) has connections to other areas of mathematics such as algebraic geometry, topology, and representation theory.